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# AP Calc BC in a week - Part 5

## Introduction

Yay! Last day! Today is:

• 10 Sequences and Series

Let's start!

This section is long so I'm going to have to split it up quite a bit...

## 10 Sequences and Series

### What is a(n) (Infinite) Sequence?

An infinite sequence is a function which has the natural numbers as the domain (not including 0). It is denoted as $$a_n$$.

An example of a sequence is $$a_n=\frac{1}{n}$$, which results in the sequence $$\frac{1}{1},\frac{1}{2},\frac{1}{3}\cdots\frac{1}{n}\cdots$$. $$\frac{1}{n}$$ is often called the nth term of this sequence.

I suppose this is to contrast sequences from the other functions that we deal with, whose domain consist of all the real numbers.

A sequence $$a$$ converges to a finite number $$L$$ if $$\lim_{n\to\infty}a_n=L$$. If the limit does not exist then the sequence diverges.

tl;dr A sequence is a (function) list of numbers, and if the limit ( $$n\to\infty$$ ) of its function is $$L$$, then it converges to $$L$$.

### What is a(n) (Infinite) Series?

An infinite series is the sum of an infinite sequence. $$\sum_{k=1}^\infty a_k = a_1 + a_2 + a_3 + \cdots + a_n + \cdots$$ Each element in the sum is a term, with $$a_n$$ being the nth term, like in the sequence.

The series $$\sum_{k=1}^\infty a_k$$ can also be written as $$\sum a_k$$ or $$\sum a_n$$.

tl;dr Add all the numbers in a sequence = series

### Types of Series

p-series is the series of form $$\sum_{k=1}^\infty \frac{1}{k^p}$$, where $$p$$ is a constant.

Harmonic series is the p-series where $$p=1$$, ie $$\sum_{k=1}^\infty \frac{1}{k}$$

Geometric series is the sum of a first term $$a$$ and terms of a common ratio $$r$$ : $$\sum_{k=1}^\infty ar^{k-1}$$

### Convergence of a Series

If there exists a finite number $$S$$ such that $$\lim_{n\to\infty}\sum_{k=1}^ne a_k = S$$, then the inifinite series is said to converge to $$S$$.

In this case, $$\sum_{k=1}^\infty = S$$.

tl;dr If you keep adding and it approaches a number, then the series converges to that number

There are a couple examples of proving that series are Divergent/convergent, but... I don't think I can reproduce these with an unseen problem, since these involve finding limits of summations, so... hopefully they can be reasoned through if they do come out.

### Theorems of Convergence of Divergence

#### If $$\sum a_n$$ converges, then $$\lim_{n\to\infty}a_n=0$$

• By contraposition, this also means that if $$\lim_{k\to\infty}a_n\neq 0$$, $$\sum a_n$$ diverges
• The converse is not true; if the limit is zero, the summation may or may not converge (if you know your truth tables this won't be a problem)
• Note that this means that the terms go to 0, not the series

#### You can add a finite number of terms to a summation without affecting its convergence/divergence

• This means that $$\sum_{k=1}^\infty a_k$$ and $$\sum_{k=m}^\infty a_k$$ both converge or both diverge

#### A series can be multiplied by a nonzero constant without affecting its convergence/divergence

• This means that $$\sum_{k=1}^\infty a_k$$ and $$c\sum_{k=1}^\infty a_k$$ both converge or both diverge

#### If two series converge, the sum of the two series converges

• If $$\sum a_n$$ and $$\sum b_n$$ both converge, $$\sum (a_n+b_n)$$ converges

## Conclusion

Uhh okay I've had this buffer open for too long I need to publish this prematurely, plus the post will become way too long anyway.

Chapter 10 is longer than I expected... I'll have to break it up. Next time we'll be doing Tests for convergences, then after that we'll do the Power series.

2016-03-29 Paul Elder

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